Deterministic 1-k routing on meshes with applications to worm-hole routing

In $1$-$k$ routing each of the $n^2$ processing units of an $n \times n$ mesh connected computer initially holds $1$ packet which must be routed such that any processor is the destination of at most $k$ packets. This problem reflects practical desire for routing better than the popular routing of permutations. $1$-$k$ routing also has implications for hot-potato worm-hole routing, which is of great importance for real world systems. We present a near-optimal deterministic algorithm running in \sqrt{k} \cdot n / 2 + \go{n} steps. We give a second algorithm with slightly worse routing time but working queue size three. Applying this algorithm considerably reduces the routing time of hot-potato worm-hole routing. Non-trivial extensions are given to the general $l$-$k$ routing problem and for routing on higher dimensional meshes. Finally we show that $k$-$k$ routing can be performed in \go{k \cdot n} steps with working queue size four. Hereby the hot-potato worm-hole routing problem can be solved in \go{k^{3/2} \cdot n} steps

Anomalous biased diffusion in networks

We study diffusion with a bias towards a target node in networks. This problem is relevant to efficient routing strategies in emerging communication networks like optical networks. Bias is represented by a probability $p$ of the packet/particle to travel at every hop towards a site which is along the shortest path to the target node. We investigate the scaling of the mean first passage time (MFPT) with the size of the network. We find by using theoretical analysis and computer simulations that for Random Regular (RR) and Erd\H{o}s-R\'{e}nyi (ER) networks, there exists a threshold probability, $p_{th}$, such that for $p<p_{th}$ the MFPT scales anomalously as $N^\alpha$, where $N$ is the number of nodes, and $\alpha$ depends on $p$. For $p>p_{th}$ the MFPT scales logarithmically with $N$. The threshold value $p_{th}$ of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes. An exact solution for every value of $p$ is given for the scaling of the MFPT in RR networks. The regime transition is also observed for the second moment of the probability distribution function, the standard deviation.Comment: 13 Pages, To appear in PR

Reinforcing Reachable Routes

This paper studies the evaluation of routing algorithms from the perspective of reachability routing, where the goal is to determine all paths between a sender and a receiver. Reachability routing is becoming relevant with the changing dynamics of the Internet and the emergence of low-bandwidth wireless/ad-hoc networks. We make the case for reinforcement learning as the framework of choice to realize reachability routing, within the confines of the current Internet infrastructure. The setting of the reinforcement learning problem offers several advantages,including loop resolution, multi-path forwarding capability, cost-sensitive routing, and minimizing state overhead, while maintaining the incremental spirit of current backbone routing algorithms. We identify research issues in reinforcement learning applied to the reachability routing problem to achieve a fluid and robust backbone routing framework. This paper also presents the design, implementation and evaluation of a new reachability routing algorithm that uses a model-based approach to achieve cost-sensitive multi-path forwarding; performance assessment of the algorithm in various troublesome topologies shows consistently superior performance over classical reinforcement learning algorithms. The paper is targeted toward practitioners seeking to implement a reachability routing algorithm